Marzo and Seip [1]} showed that \(P_+\)  is a contractive operator from \(L^q (\)  to \(H^\frac{4q}{q + 2}\)  if \(2 \le q \le \infty \) , and using duality arguments the same can be said from \(L^q (\)  to \(H^\frac{2q}{4-q}\)  if \(\frac{4}{3} \le q \le 2\)  [2]}. However, the norm of \(P_+ : L^q ( \rightarrow X\)  is not known in general if \(X\)  is a space of analytic functions containing \(H^q\) . In the case that \(X\)  is a Hardy space, we can find in the literature the conjecture below.
